3.862 \(\int \frac{1}{\left (a+b x^4\right ) \sqrt{c x^2+d \sqrt{a+b x^4}}} \, dx\)

Optimal. Leaf size=40 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}+c x^2}}\right )}{a \sqrt{c}} \]

[Out]

ArcTanh[(Sqrt[c]*x)/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]]/(a*Sqrt[c])

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Rubi [A]  time = 0.220544, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}+c x^2}}\right )}{a \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)*Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]),x]

[Out]

ArcTanh[(Sqrt[c]*x)/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]]/(a*Sqrt[c])

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Rubi in Sympy [A]  time = 6.40103, size = 34, normalized size = 0.85 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{c x^{2} + d \sqrt{a + b x^{4}}}} \right )}}{a \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)/(c*x**2+d*(b*x**4+a)**(1/2))**(1/2),x)

[Out]

atanh(sqrt(c)*x/sqrt(c*x**2 + d*sqrt(a + b*x**4)))/(a*sqrt(c))

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Mathematica [A]  time = 0.16237, size = 50, normalized size = 1.25 \[ \frac{\sqrt{-\frac{1}{c}} \cot ^{-1}\left (\frac{\sqrt{-\frac{1}{c}} \sqrt{d \sqrt{a+b x^4}+c x^2}}{x}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^4)*Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]),x]

[Out]

(Sqrt[-c^(-1)]*ArcCot[(Sqrt[-c^(-1)]*Sqrt[c*x^2 + d*Sqrt[a + b*x^4]])/x])/a

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{4}+a}{\frac{1}{\sqrt{c{x}^{2}+d\sqrt{b{x}^{4}+a}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x)

[Out]

int(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{c x^{2} + \sqrt{b x^{4} + a} d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{4}\right ) \sqrt{c x^{2} + d \sqrt{a + b x^{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**4+a)/(c*x**2+d*(b*x**4+a)**(1/2))**(1/2),x)

[Out]

Integral(1/((a + b*x**4)*sqrt(c*x**2 + d*sqrt(a + b*x**4))), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{c x^{2} + \sqrt{b x^{4} + a} d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)), x)